Method, program product and apparatus for predicting line width roughness and resist pattern failure and the use thereof in a lithography simulation process

ABSTRACT

A method of generating a model for simulating the imaging performance of an optical imaging system. The method includes the steps of defining the optical imaging system and a process to be utilized by the optical imaging system; defining a first model representing the imaging performance of the optical imaging system and the process, and calibrating the model, where the first model generates values corresponding to a latent image slope. The method further includes the step of defining a second model for estimating a line width roughness of a feature to be imaged, where the second model utilizes the latent image slope values to estimate the line width roughness.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.11/979,838, filed on Nov. 8, 2007, which claims priority to U.S.Provisional Application Ser. No. 60/857,495, filed on Nov. 8, 2006, thecontents of which are incorporated by reference herein in theirentirety.

TECHNICAL FILED

This disclosure relates generally to lithographic simulation tools. Moreparticularly, it relates to a method for modeling and/or predicting linewidth roughness and resist pattern failure in a lithography process anda method for utilizing the same in a lithographic simulation process.

BACKGROUND

Lithographic apparatus can be used, for example, in the manufacture ofintegrated circuits (ICs). In such a case, the mask may contain acircuit pattern corresponding to an individual layer of the IC, and thispattern can be imaged onto a target portion (e.g., comprising one ormore dies) on a substrate (silicon wafer) that has been coated with alayer of radiation-sensitive material (resist). In general, a singlewafer will contain a whole network of adjacent target portions that aresuccessively irradiated via the projection system, one at a time. In onetype of lithographic projection apparatus, each target portion isirradiated by exposing the entire mask pattern onto the target portionin one go; such an apparatus is commonly referred to as a wafer stepper.In an alternative apparatus, commonly referred to as a step-and-scanapparatus, each target portion is irradiated by progressively scanningthe mask pattern under the projection beam in a given referencedirection (the “scanning” direction) while synchronously scanning thesubstrate table parallel or anti-parallel to this direction. Since, ingeneral, the projection system will have a magnification factor M(generally >1), the speed V at which the substrate table is scanned willbe a factor M times that at which the mask table is scanned. Moreinformation with regard to lithographic devices as described herein canbe gleaned, for example, from U.S. Pat. No. 6,046,792, incorporatedherein by reference.

In a manufacturing process using a lithographic projection apparatus, amask pattern is imaged onto a substrate that is at least partiallycovered by a layer of radiation-sensitive material (resist). Prior tothis imaging step, the substrate may undergo various procedures, such aspriming, resist coating and a soft bake. After exposure, the substratemay be subjected to other procedures, such as a post-exposure bake(PEB), development, a hard bake and measurement/inspection of the imagedfeatures. This array of procedures is used as a basis to pattern anindividual layer of a device, e.g., an IC. Such a patterned layer maythen undergo various processes such as etching, ion-implantation(doping), metallization, oxidation, chemo-mechanical polishing, etc.,all intended to finish off an individual layer. If several layers arerequired, then the whole procedure, or a variant thereof, will have tobe repeated for each new layer. Eventually, an array of devices will bepresent on the substrate (wafer). These devices are then separated fromone another by a technique such as dicing or sawing, whence theindividual devices can be mounted on a carrier, connected to pins, etc.

For the sake of simplicity, the projection system may hereinafter bereferred to as the “optics;” however, this term should be broadlyinterpreted as encompassing various types of projection systems,including refractive optics, reflective optics, and catadioptricsystems, for example. The radiation system may also include componentsoperating according to any of these design types for directing, shapingor controlling the projection beam of radiation, and such components mayalso be referred to below, collectively or singularly, as a “lens.”Further, the lithographic apparatus may be of a type having two or moresubstrate tables (and/or two or more mask tables). In such “multiplestage” devices the additional tables may be used in parallel, orpreparatory steps may be carried out on one or more tables while one ormore other tables are being used for exposures. Twin stage lithographicapparatus are described, for example, in U.S. Pat. No. 5,969,441,incorporated herein by reference.

The photolithographic masks referred to above comprise geometricpatterns corresponding to the circuit components to be integrated onto asilicon wafer. The patterns used to create such masks are generatedutilizing CAD (computer-aided design) programs, this process often beingreferred to as EDA (electronic design automation). Most CAD programsfollow a set of predetermined design rules in order to create functionalmasks. These rules are set by processing and design limitations. Forexample, design rules define the space tolerance between circuit devices(such as gates, capacitors, etc.) or interconnect lines, so as to ensurethat the circuit devices or lines do not interact with one another in anundesirable way. A critical dimension of a circuit can be defined as thesmallest width of a line or hole or the smallest space between two linesor two holes. Thus, the CD determines the overall size and density ofthe designed circuit.

As the required CDs continue to decrease, photolithographic simulationbased on physical models has become an essential tool for understandingand optimizing the advanced photolithographic processes necessary fortoday's advanced circuit designs. An important aspect of such simulationprocesses is to utilize calibrated models which accurately describe theresist process (e.g., the image to be formed in the resist layerdeposited on the substrate). This is especially true in the low-klregime where the printed images in photoresist depart significantly fromthe projected optical image. Reasons for this disparity are wellunderstood and relate to diffusion of components in the photoresist andfinite dissolution contrast. As such, without calibration of thephotoresist model to match experimental data, the utility ofphotolithographic simulation processes are significantly reduced. Acharacteristic of a well calibrated model is that the criticaldimensions (CD) of printed features are accurately predicted.

Recently though it has become clear that current simulation models maybe insufficient as many low-kl processes are limited not by details ofthe measured and predicted CDs, but by other considerations such as theroughness of the printed feature and whether it actually appears on thewafer or has suffered some type of pattern failure. Line width roughness(LWR), which is defined as the 3σ variation in the measured featurewidth, has received significant attention because as feature sizes andkl simultaneously shrink, this width variation can be an appreciablefraction of the total CD budget, and device performance can becompromised as a result of LWR.

Currently, however, generally speaking LWR and pattern failure cannot bereadily predicted by standard photolithographic simulation tools, and infact, their prediction is typically not even addressed. Models forcalculating microscopic roughness, specifically line width roughness(LWR) have appeared in the literature, but these known techniques areslow and complex and cannot be easily incorporated into the standardmethod of working with simulators, and therefore do not offer apractical solution.

Accordingly, there exists a need for a method for predicting both LWRand pattern failure that is both practical and efficient and which canbe readily implemented into current photolithography simulationprocesses.

SUMMARY

In view of the foregoing deficiencies in the prior art imaging models,the present invention provides a method that for a given resist processallows for both LWR and pattern failure to be predicted by an empiricalmodel, which uses image metrics available from a lithographic simulationprogram. The method is particularly useful because it allows thepredictions of LWR and pattern failure to be directly output by standardlithographic simulation programs without any new or complexcalculations.

More specifically, the present invention relates to a method ofgenerating a model for simulating the imaging performance of an opticalimaging system. The method includes the steps of defining the opticalimaging system and a process to be utilized by the optical imagingsystem; defining a first model representing the imaging performance ofthe optical imaging system and the process, and calibrating the model,where the first model generates values corresponding to a latent imageslope. Specifically, first the first model is calibrated in the typicalmanner so as to match the experimental critical dimensions (CDs). As anintermediate step in this process, values for the latent image slope arealso generated. Then, the method further includes the step of defining asecond model for estimating a line width roughness of a feature to beimaged, where the second model utilizes the latent image slope values toestimate the line width roughness.

In a second embodiment, the present invention relates to a method ofpredicting the occurrence of pattern failure. The method of thisembodiment includes the steps of defining said optical imaging systemand a process to be utilized by the optical imaging system; defining afirst model representing the imaging performance of the optical imagingsystem and the process, and calibrating said model, where the firstmodel generates, for example, CDs to match experiment and latent imageslope values as intermediates. The method further includes the step ofdefining a second model for estimating a pattern failure regarding afeature to be imaged, where the second model utilizes the latent imageslope values to define parameters which are utilized to predict whenpattern failure of a given feature will occur.

As explained in more detail below, the method of the present inventionprovides significant advantages over the prior art. Most importantly,the present invention provides a simulation model which can accuratelyestimate LWR and predict pattern failure, and which is readilyimplemented utilizing existing simulation models. As a result, a moreaccurate imaging model and simulation result is produced, which benefitsall applications utilizing such simulation processes. For example,including pattern failure and LWR information into the analysis ofpattern printing allows an IC manufacturer to better identify areaswhere the device might fail and allow the manufacturer to improve themask or scanner conditions to avoid costly delays.

Although specific reference may be made in this text to the use of theinvention in the manufacture of ICs, it should be explicitly understoodthat the invention has many other possible applications. For example, itmay be employed in the manufacture of integrated optical systems,guidance and detection patterns for magnetic domain memories,liquid-crystal display panels, thin-film magnetic heads, etc. Theskilled artisan will appreciate that, in the context of such alternativeapplications, any use of the terms “reticle,” “wafer” or “die” in thistext should be considered as being replaced by the more general terms“mask,” “substrate” and “target portion,” respectively. Althoughspecific reference may be made in this text to the use of the inventionin the manufacture of ICs, it should be explicitly understood that theinvention has many other possible applications. For example, it may beemployed in conjunction with the manufacture of integrated opticalsystems, guidance and detection patterns for magnetic domain memories,liquid-crystal display panels, thin-film magnetic heads, etc. Theskilled artisan will appreciate that, in the context of such alternativeapplications, any use of the terms “reticle”, “wafer” or “die” in thistext should be considered as being replaced by the more general terms“mask”, “substrate” and “target portion”, respectively.

In the present document, the terms “radiation” and “beam” are used toencompass all types of electromagnetic radiation, including ultravioletradiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) andEUV (extreme ultra-violet radiation, e.g. having a wavelength in therange 5-20 nm).

The term mask as employed in this text may be broadly interpreted asreferring to generic patterning means that can be used to endow anincoming radiation beam with a patterned cross-section, corresponding toa pattern that is to be created in a target portion of the substrate;the term “light valve” can also be used in this context. Besides theclassic mask (transmissive or reflective; binary, phase-shifting,hybrid, etc.), examples of other such patterning means include:

-   -   a programmable mirror array. An example of such a device is a        matrix-addressable surface having a viscoelastic control layer        and a reflective surface. The basic principle behind such an        apparatus is that (for example) addressed areas of the        reflective surface reflect incident light as diffracted light,        whereas unaddressed areas reflect incident light as undiffracted        light. Using an appropriate filter, the said undiffracted light        can be filtered out of the reflected beam, leaving only the        diffracted light behind; in this manner, the beam becomes        patterned according to the addressing pattern of the        matrix-addressable surface. The required matrix addressing can        be performed using suitable electronic means. More information        on such mirror arrays can be gleaned, for example, from U.S.        Pat. Nos. 5,296,891 and 5,523,193, which are incorporated herein        by reference.    -   a programmable LCD array. An example of such a construction is        given in U.S. Pat. No. 5,229,872, which is incorporated herein        by reference.

The invention itself, together with further objects and advantages, canbe better understood by reference to the following detailed descriptionand the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates how image contrast and latent image contrast varywith dose.

FIGS. 2a and 2b illustrate resulting latent image slope valuesassociated with experimental results for two different resist processes.

FIG. 3 illustrates data for the special process as mentioned in FIGS. 2aand 2b , and a curve produced by the LWR model which results fromfitting the model parameters to the experimental data.

FIG. 4 shows experimental measurement of LWR for 50 nm 1:1 lines imagedwith 1.2 NA and 0.94/0.74 annular illumination through dose and focus.

FIG. 5a shows a fit of the LIslope values obtained from the resist modelto the LWR measurements using the empirical 3-parameter LWR equation.FIG. 5b illustrates an overlay of the LWR predictions and measurementsthrough dose and focus.

FIGS. 6(a)-6(f) illustrate an example of the how the LWR model can beutilized in a standard lithographic simulation process.

FIG. 7 illustrates another example of the how the LWR model can beutilized in a standard lithographic simulation process.

FIGS. 8a and 8b illustrate a comparison of process windows in the formof EL vs. DOF curves for experimental data and model predictions.

FIG. 9 is an exemplary SEM image illustrating an example of patternfailure.

FIG. 10 illustrates experimental EL vs. DOF results for 55 nm 1:1 linesprinted with 0.93 NA and dipole illumination.

FIGS. 11(a)-11(c) illustrate simulation results regarding predicting CDsthrough focus and dose.

FIGS. 12(a)-12(c) illustrate an application of the three parameterpattern failure model of the present invention.

FIGS. 13(a)-13(b) illustrate another application of the three parameterpattern failure model of the present invention.

FIGS. 14(a)-14(b) illustrate yet another application of the threeparameter pattern failure model of the present invention.

FIG. 15 illustrates experimental data including SEM images at the edgeof focus and illustrates that pattern failure can involve both patterncollapse and space closing.

FIGS. 16a and 16b illustrate a comparison between the performance of thethree parameter pattern failure model and the five parameter patternfailure model of the present invention.

FIGS. 17a-17c illustrate an exemplary application of the five parameterpattern failure model

FIG. 18 illustrates an exemplary flowchart regarding the generation ofthe LWR model and the pattern failure models.

FIG. 19 is a block diagram that illustrates a computer system 100 whichcan assist in the generation of a LWR model and the pattern failuremodels of the present invention.

FIG. 20 schematically depicts an exemplary lithographic projectionapparatus that could be the basis of the imaging model to be developedand calibrated in accordance with the process of the present invention.

DETAILED DESCRIPTION

In accordance with the present invention, and as explained in furtherdetail below, the preferred image metric used for modeling and/orpredicting both LWR and pattern failure is the slope of the latent image(LI_(slope)) in the resist film at the edge of the developed resistfeature. LI_(slope) is a direct description of the quality of the imagein the resist prior to development and it is strongly related to thequality of the developed resist image. The LI_(slope) quantity/value isreadily available from the typical simulator (such as those offered byASML) using a standard calibrated model because for any dose, focus,illumination, mask, NA, polarization, etc., condition the simulatorreturns a CD prediction, which defines the feature edge, and retains inan intermediate step, the latent image slope (LI_(slope)) correspondingwith the given position and imaging conditions. It is noted that othersimilar metrics may also provide useful predictors for LWR and patternfailure, for example, but not limited to, latent image NILS (normalizedimage log slope), latent image ILS (image log slope), and the simpler,analogous optical image metrics. As noted, LWR is dependent on imagecontrast (or NILS) and more precisely the image slope at the featureedge.

It is noted that there are different mathematical formulae that can beselected to relate the LI_(slope) to experimental observations (i.e.,actual imaging results) of LWR and pattern failure. The followingformula have been found to hold quite well over an appreciable range.More specifically, the formula utilized in the exemplary embodimentsdisclosed herein are:LWR=a·(LI_(slope))^(b) =c,where a, b, and c are empirical constants obtained by fitting theconstants to actual experimental data such that the LWR numbers producedby the formula match the experimental/actual LWR results with somepredefined error criteria.

With regard to predicting pattern failure, it is first noted thatpattern failure can occur in at least two different instances. The firstis when the aspect ratio of the feature to be imaged becomes to high. Insuch instances, the feature may literally fall over, for example, due tocapillary forces from adjacent features during the develop/rinseprocess. The second is when the features to be imaged are too closetogether and start to merge (this is referred to as “scumming”). Thissort of pattern failure may be related to excessive LWR and smallspatial separation leading to “bridging” between adjacent lines.

For pattern failure predictions, two different formulas/models have beendeveloped, a three parameter model (which his intended to detect failuredue to features falling over), and a five parameter model (which isintended to detect failures due to features falling over and due toscumming).

With regard to the three parameter model, for a given (i.e., fixed)resist process, the line feature is good (i.e. has a CD as predicted bylithographic simulator, and the feature is actually measurable on thewafer), if:

-   -   1. CD>CD_(min)    -   2. LI_(slope)>LI_(slopemin)    -   3. (CD-CD_(min))*(LI_(slope)-LI_(slope min))>C_(ct)*CD_(min)        (3-parameter model)        The three fitting constants of the three parameter model are        CD_(min), LI_(slopemin), and C_(ct). The physical interpretation        of these parameters are as follows: 1) if the CD is too small        (<CD_(min)) the feature will fall over and be washed off the        wafer in the develop step, 2) if the chemical contrast is too        low (LI_(slopemin)), the feature will not cleanly develop, even        though, the mathematics of the simulator predict it will, and 3)        when near the critical CD or the minimum LIslope there is an        interaction, or cross-term between the two and the pattern        actually fails even though both individual criteria are OK;        Cct*CDmin is the coefficient for this cross-term. Similar to        determining the constants for the LWR model, the foregoing        parameters, CDmin, LIslopemin, and Cct, are obtained by        determining the values of these constants at which the criteria        predicting pattern failure actually match the experimental data.    -   For the 5-parameter model, the pattern is good if:    -   1. CD>CD_(min)    -   2. LI_(slope)>LI_(slope min) (if CD_(line)>CD_(space))    -   3. (CD-CD_(min))*(LI_(slope)-LI_(slope min))>C_(ct)*CD_(min)    -   4. pitch-CD>Space_(min)    -   5. LI_(slope)>LI_(slope min2) (if CD_(space)e>CD_(line))        (5-parameter model)        The two additional fitting constants are Space_(min) and        LI_(slopemin2) where the model now also detects pattern failures        associated with spaces closing (i.e., scumming) not just lines        falling over. This is particularly useful for dense features        where the spaces and lines are comparable in size and pattern        failure can occur either from lines falling over or spaces        prematurely closing up.

A more detailed explanation of how the foregoing models predict both LWRand pattern failure is now provided. FIG. 1 illustrates, for a typicalmodern imaging problem, how image contrast and latent image contrastvary with dose. As further explained below, LWR also varies with dose.As such, FIG. 1 indicates that latent image metrics are preferable toimage metrics in predicting LWR.

FIGS. 2a and 2b illustrates experimental results for two differentresist processes: 1) a “standard” process and 2) a lower LWR processreferred to as “special”, with two different imaging problems: 1) 55 nm1:1 lines with dipole illumination, and 2) 65 nm 1:1 lines with annularillumination, both using 193 nm light and a 0.93 NA lens. FIG. 2a plotsthe measured LWR values vs. the latent image contrast calculated using acalibrated model for each data point. Though for each condition (resistprocess and feature type/illumination) the plots are monotonic, thereare four distinct curves. In contrast, FIG. 2b , which plots LWR vs. thelatent image slope taken at the developed feature edge, separates intotwo curves, one for each resist process. As such, FIG. 2b indicates thatLIslope at the feature edge is a superior predictor for LWR associatedwith a particular resist process than is LIcontrast. It is also shownthat the standard process consistently gives LWR 2 nm higher than thespecial process.

FIG. 3 illustrates data for the special process as mentioned in FIGS. 2aand 2b but also combines data taken at 1.2 NA for 42 nm 1:1 lines and 45nm 1:1 lines produced with dipole illumination. A minor model adjustmentwas performed to also match the experimental dose latitude measurements,and the result is smooth monotonic behavior, which is fitted to thepower series equation for predicting LWR set forth above. As such, it isclear that for other imaging problems with this resist process, the LWRcan be sufficiently predicted by calculating the LIslope at the featureedge and using the foregoing equation having the constants fitted to thegiven resist process. In the given example, the equation becomes:LWR=90.9·(LI_(slope))^(−0.85)

A further check of the utility of this prediction is shown in Table 1where the model is applied including a 2 nm offset because the standardprocess is used. Referring to Table 1, data for two different featuretypes (55 nm 1:1 and 60 nm 1:1) is compared at best focus and dose, bothwith and without polarized illumination. The predictions are reasonablyclose to the experimental values, and illustrates that polarizationgives better LWR and a better latent image.

A further indication of the sufficiency of the model is shown in FIG. 4and FIGS. 5a and 5b . FIG. 4 shows experimental measurement of LWR for50 nm 1:1 lines imaged with 1.2 NA and 0.94/0.74 annular illuminationthrough dose and focus. It is shown that LWR increases as the image isdefocused and with underexposure. FIG. 5a shows a smooth curve resultingfrom the three indicated coefficients and the modeled LIslope values,obtained from the resist model, overlaid on the experimental LWRmeasurements. It is shown that the model substantially reproduces theexpected data, though clearly there is some noise in the experimentaldata, typical of LWR measurements. FIG. 5b overlays the LWR predictionsand measurements through dose and focus. Again, the agreement issatisfactory, further validating the LWR model of the presentembodiment.

An example of the way the LWR model can be included in standardlithographic simulations is shown in FIGS. 6(a)-(c) and 6(d)-(f). FIGS.6(a)-(c) consider the case of polarized illumination and illustratesplots from the model which indicate the focus:exposure process windowbased on CD (FIG. 6a ), the LWR contours through dose and focus (FIG. 6b), and the overlapping process window where CD is within 10% of thetarget and LWR is less than some preset limit (8.5 used here) (FIG. 6c). FIGS. 6(d)-(f) illustrate the analogous results without polarization.Upon review of the figures, it is shown that the overlapping processwindow is much smaller without polarization, primarily because of thehigher LWR.

FIG. 7 illustrates another example of the utility of the LWR model in asimulation program. In this case, the LWR model returns a numericalvalue which can be synthesized into a top-down view, similar to whatwould be shown in a SEM measurement of the corresponding resist feature.It is noted that noise generators can be utilized to provide the desiredcosmetic result or more complex and rigorous based on experimental powerspectral density measurements, or similar treatments of the roughnesscan be used.

As noted above, pattern failure is also a major problem and difficultaspect for accurate photolithographic modeling. An example of theshortcomings of standard models is shown in FIGS. 8a and 8b , whichcompares process windows in the form of EL vs. DOF curves forexperimental data and model predictions. The results are for printing 60nm 1:1 lines with 0.93 NA and dipole illumination with threedifferences: 1) dry system unpolarized, 2) dry system polarized, and 3)water immersion system polarized. As shown in FIG. 8b , the calibratedmodel predicts the dose latitude and general shapes of the curvesreasonably well with one major exception—all of the experimental curvesshown in FIG. 8a (which represent the actual imaging result) areabruptly truncated due to pattern failure, while the simulation resultpredicts much higher DOF. It is this discrepancy that is addressed bythe pattern failure model of the present invention.

FIG. 9 examines the nature of the experimental truncation. Referring toFIG. 9, it is shown that the resist images at 0.30 and −0.30 u focussetting (as well as all in between, not shown) are fine and will print,and CDs are measurable and near the target to print 1:1 lines andspaces. However, the images just 0.05 u further out show a problem withthe lines falling over and the pattern is failing, i.e. not usable forproduction and cannot even be reliably measured. With regard to thepatterns that remain, the CD is good as is typically predicted by thesimulation program, but the pattern has failed, which is not predictedby current simulation models.

Examination of several experimental datasets led to the conclusion thatpattern failure in dense lines for a particular resist process can bepredicted reasonably well by simply invoking two criteria: 1) a CDsmaller than some minimum value would not be resolved without failing,and 2) below some critical LIslope at the developed feature edge, thepattern would also fail. The former criteria is expected as it is knownthat beyond some critical aspect ratio, capillary forces in the developand rinse process causes lines to fall over. However, the LI_(slope)constraint is less intuitive; the simulation model indicates that thefeature should be printed with an acceptable CD (often well above thecritical minimum CD), and the lines which do not fail are in agreementwith this CD, but some other mechanical failure occurs when theLI_(slope) quality falls below a critical value.

Examples of the foregoing are provided in FIGS. 10 and 11(a)-(c). FIG.10 illustrates the experimental EL vs. DOF results for 55 nm 1:1 linesprinted with 0.93 NA and dipole illumination. The abrupt cutoff in focuscan be seen and it occurs much earlier for unpolarized illumination thanpolarized. In contrast. FIG. 11a illustrates the results of a simplesimulation process predicting CDs through focus and dose for theunpolarized case. As is clear, the expected DOF is almost twice as highas the experimental/actual value. FIG. 11b truncates the simulationresults with the following two rules based on the empirical data: for afeature to be good and measurable the CDmin>48 nm and LIslope>14. Thephysical interpretation is that CDs less than 48 nm (overexposed case)fall over, while features with critically poor latent image quality(defocused case) also fail. With these criteria, the simulated DOFmatches experiment. FIG. 11c shows the analogous result for thepolarized case, and again the simulation is shown to match theexperimental DOF in FIG. 10.

While this 2-parameter pattern failure model is quite useful, FIGS.12a-12c show that additional refinement by adding a 3^(rd) parameterfurther improves the match between the simulated result and theexperimental or actual result. Specifically, FIG. 12a shows experimentalFEM measurements for 60 nm 1:1 lines produced with dipole illuminationat 0.93 NA in a water immersion system. FIG. 12b shows the best match ofsimulation when the simple 2-parameter model is applied (which isobtained by fitting the 2-parameter model to the experimental data). Aproblem area exists where the model predicts more good points whenoverexposed and defocused than actually appear on the wafer. Thesepoints have the characteristic that the CDs are small, approaching butstill above the critical CDmin, and also, in defocus, the latent imageslope is degraded though still above the critical value. By adding a3^(rd) criteria that allows for failure above the critical limits butwhen near failure for both, helps solve this problem as shown in FIG.12c , where the agreement between the simulated result and theexperimental result is improved. This three parameter model is definedas follows:

-   -   1. CD>CD_(min)    -   2. LI_(slope)>LI_(slopemin)    -   3. (CD-CD_(min))*(LI_(slope)-LI_(slope min))>C_(ct)*CD_(min)        (3-parameter model)

It is noted that the values of CD_(min), LI_(slope min), and C_(ct) aredetermined by a comparison of the experimental data and the modelpredictions so as to determine the parameters values such that thesimulation result most closely matches the experimental data with aminimal number of errors (i.e., predicting a pattern failure where noneoccurs, or vise versa). Another example of the ability of the model tosuccessfully estimate the actual experimental results is shown in FIGS.13a and 13b , which uses a different resist process and examines 45 nm1:1 lines with 1.2 NA dipole illumination. The experimental DOF is ˜0.5u while the simple matched simulation prediction is 0.7 u. When the3-parameter pattern failure criterion is added then the DOF match isimproved as is the predicted shape of the measured Bossung plot.

Yet another example is shown in FIGS. 14a and 14b . It is noted thatFIG. 14a is a repeat of FIG. 4b . In this example, the experimental data(the plotted discrete points) and the simulation results including the3-parameter failure criterion are overlayed. It is shown that theagreement between the simulated result and the experimental result isquite good. The process is for 50 nm 1:1 lines produced with 1.2 NA and0.94/0.74 annular illumination and the best model parameters are:

Cdmin LI slope min Cct 31 13 1.9

FIG. 14b shows the simulation result without including pattern failureand its over-prediction of the process window is clear. It is noted thatthis dataset is the same as used for the LWR analysis of FIG. 5 and so,taken together, it is shown that including these empirical models it ispossible to 1) predict CDs, 2) predict LWR, and 3) predict patternfailure.

Further refinement of the pattern failure model is also possible. FIG.15 shows experimental data including SEM images at the edge of focuswhich makes clear that pattern failure can involve both pattern collapsefor small lines and also “premature space closing” for small spaces. Ina second embodiment of the pattern failure model, two more terms can beadded to the first pattern failure model discussed above to include thiseffect. In such a case, the model becomes:

-   -   1. CD>CD_(min)    -   2. LI_(slope)>LI_(Slope min) (if CD_(line)>CD_(space))    -   3. (CD-CD_(min))*(LI_(slope)-LI_(slope min))>C_(ct)*CD_(min)    -   4. pitch-CD>Space_(min)    -   5. LI_(slope)>LI_(slope min2) (if CD_(space)>CD_(line))        (5-parameter model)        Once again, the values of the model CD_(min), LI_(slope min),        C_(ct), Space_(min), and LI_(slope min2) are determined by a        comparison of the experimental data and the parameter values so        as to determine the parameters values such that the simulation        result most closely matches the experimental data with the        minimal number of errors (i.e., predicting a pattern failure        where none occurs, or vise versa).

FIGS. 16a and 16b show the utility of the model. When the 5 parametermodel is calibrated to match 50 nm 1:1 annular data, its prediction for42 nm 1:1 dipole data is much better than the prediction of the3-parameter model. FIG. 16b which used the 3-parameter model predictsmuch better performance in underexposure (small spaces) than is observedby the experiment. FIG. 16a shows that the 5-parameter model reducesthis discrepancy.

FIGS. 17a-17c make this point more clearly. Here the models are directlyfitted to the 42 nm 1:1 data. FIG. 17a shows a good match with the5-parameter model while FIG. 17b shows a worse match, particularly inunderexposure for the 3-parameter model. FIG. 17c shows the predictionwith no pattern failure considered.

FIG. 18 illustrates an exemplary flowchart regarding the generation ofthe LWR model and the pattern failure model noted above. In the firststep of the process 180, actual imaging results are obtained for thegiven resist process to be utilized. The experimental data points aretaken through focus and dose. Typically, the measured data is theresulting CD, however, other measurements are also possible. As notedabove, the foregoing models are resist process dependent and thecalibration of the model (i.e., determination of the model parameters)should be re-performed if the resist process is modified. The next step(Step 182) is to calibrate the given simulation model being utilized (itis noted that any prior art simulation model in which the value of theLI_(slope) is available as an intermediary value may be utilized). Thisstep corresponds to a typical simulation model calibration process. Inthe next step (Step 184), using the calibrated resist model, determinethe simulated CD values and corresponding the LIslope values generatedby the simulation model which correspond to the experimental (i.e.,actual) imaging results. Then (Step 186), the three parameters in theLWR model are determined by fitting the three parameters of the LWRmodel to the experimental data such that the simulation results of theLWR model most closely match the experimental results. This can beaccomplished for example, by minimizing the sum of the squares of themodeled and experimental LWR differences. Thereafter, in the final step(Step 188), either the three parameters or five parameters of thepattern failure model are determined by a comparison of the experimentaldata and the parameter values so as to determine the parameters valuessuch that the simulation result most closely matches the experimentaldata with the minimal number of errors (i.e., predicting a patternfailure where none occurs, or vise versa).

In a variation of the foregoing method of determining the parametersassociated with the pattern failure model, it is also possible tointroduce an additional metric to be minimized, which is:metric=(S ⁺ E+E ⁺ S ⁻)+|S ⁺ E ⁻ −E ⁺ S ⁻|,wherein S⁺E⁻ indicates that the pattern is good in simulation, but failsin experimental result, and E⁺S⁻ indicates that the pattern is good inthe experimental results but fails in simulation. The purpose of thismetric is to balance the two types of possible errors when fitting theparameters of the model to the experimental data.

It is noted that while specific mathematical functions have beendisclosed above, other formula having additional parameters may also beutilized. Further, as opposed to calibrating the parameters of the LWRmodel and pattern failure model over variations of focus and dose, a onepoint calibration can be utilized (for example, at best focus and bestdose). Of course, other variations are also possible.

FIG. 19 is a block diagram that illustrates a computer system 100 whichcan assist in the generation and use of the LWR and pattern failuremodels in accordance with the present invention. Computer system 100includes a bus 102 or other communication mechanism for communicatinginformation, and a processor 104 coupled with bus 102 for processinginformation. Computer system 100 also includes a main memory 106, suchas a random access memory (RAM) or other dynamic storage device, coupledto bus 102 for storing information and instructions to be executed byprocessor 104. Main memory 106 also may be used for storing temporaryvariables or other intermediate information during execution ofinstructions to be executed by processor 104. Computer system 100further includes a read only memory (ROM) 108 or other static storagedevice coupled to bus 102 for storing static information andinstructions for processor 104. A storage device 110, such as a magneticdisk or optical disk, is provided and coupled to bus 102 for storinginformation and instructions.

Computer system 100 may be coupled via bus 102 to a display 112, such asa cathode ray tube (CRT) or flat panel or touch panel display fordisplaying information to a computer user. An input device 114,including alphanumeric and other keys, is coupled to bus 102 forcommunicating information and command selections to processor 104.Another type of user input device is cursor control 116, such as amouse, a trackball, or cursor direction keys for communicating directioninformation and command selections to processor 104 and for controllingcursor movement on display 112. This input device typically has twodegrees of freedom in two axes, a first axis (e.g., x) and a second axis(e.g., y), that allows the device to specify positions in a plane. Atouch panel (screen) display may also be used as an input device.

Determination and calibration of the models may be performed by computersystem 100 in response to processor 104 executing one or more sequencesof one or more instructions contained in main memory 106. Suchinstructions may be read into main memory 106 from anothercomputer-readable medium, such as storage device 110. Execution of thesequences of instructions contained in main memory 106 causes processor104 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 106. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions to implement the invention.Thus, embodiments of the invention are not limited to any specificcombination of hardware circuitry and software.

The term “computer-readable medium” as used herein refers to any mediumthat participates in providing instructions to processor 104 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media, volatile media, and transmission media.Non-volatile media include, for example, optical or magnetic disks, suchas storage device 110. Volatile media include dynamic memory, such asmain memory 106. Transmission media include coaxial cables, copper wireand fiber optics, including the wires that comprise bus 102.Transmission media can also take the form of acoustic or light waves,such as those generated during radio frequency (RF) and infrared (IR)data communications. Common forms of computer-readable media include,for example, a floppy disk, a flexible disk, hard disk, magnetic tape,any other magnetic medium, a CD-ROM, DVD, any other optical medium,punch cards, paper tape, any other physical medium with patterns ofholes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip orcartridge, a carrier wave as described hereinafter, or any other mediumfrom which a computer can read.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 104 forexecution. For example, the instructions may initially be borne on amagnetic disk of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 100 canreceive the data on the telephone line and use an infrared transmitterto convert the data to an infrared signal. An infrared detector coupledto bus 102 can receive the data carried in the infrared signal and placethe data on bus 102. Bus 102 carries the data to main memory 106, fromwhich processor 104 retrieves and executes the instructions. Theinstructions received by main memory 106 may optionally be stored onstorage device 110 either before or after execution by processor 104.

Computer system 100 also preferably includes a communication interface118 coupled to bus 102. Communication interface 118 provides a two-waydata communication coupling to a network link 120 that is connected to alocal network 122. For example, communication interface 118 may be anintegrated services digital network (ISDN) card or a modem to provide adata communication connection to a corresponding type of telephone line.As another example, communication interface 118 may be a local areanetwork (LAN) card to provide a data communication connection to acompatible LAN. Wireless links may also be implemented. In any suchimplementation, communication interface 118 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

Network link 120 typically provides data communication through one ormore networks to other data devices. For example, network link 120 mayprovide a connection through local network 122 to a host computer 124 orto data equipment operated by an Internet Service Provider (ISP) 126.ISP 126 in turn provides data communication services through theworldwide packet data communication network, now commonly referred to asthe “Internet” 128. Local network 122 and Internet 128 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 120 and through communication interface 118, which carrythe digital data to and from computer system 100, are exemplary forms ofcarrier waves transporting the information.

Computer system 100 can send messages and receive data, includingprogram code, through the network(s), network link 120, andcommunication interface 118. In the Internet example, a server 130 mighttransmit a requested code for an application program through Internet128, ISP 126, local network 122 and communication interface 118. Inaccordance with the invention, one such downloaded application providesfor the determination of the calibration test patterns. The receivedcode may be executed by processor 104 as it is received, and/or storedin storage device 110, or other non-volatile storage for laterexecution. In this manner, computer system 100 may obtain applicationcode in the form of a carrier wave. generating a model of an opticalimaging system which incorporates a non-linear model of the resistperformance in accordance with the present invention.

FIG. 20 schematically depicts an exemplary lithographic projectionapparatus that could be the basis of the imaging model and CPK to bedeveloped and calibrated in accordance with the process of the presentinvention. The apparatus comprises:

-   -   a radiation system Ex, IL, for supplying a projection beam PB of        radiation. In this particular case, the radiation system also        comprises a radiation source LA;    -   a first object table (mask table) MT provided with a mask holder        for holding a mask MA (e.g., a reticle), and connected to first        positioning means for accurately positioning the mask with        respect to item PL;    -   a second object table (substrate table) WT provided with a        substrate holder for holding a substrate W (e.g., a        resist-coated silicon wafer), and connected to second        positioning means for accurately positioning the substrate with        respect to item PL;    -   a projection system (“lens”) PL (e.g., a refractive, catoptric        or catadioptric optical system) for imaging an irradiated        portion of the mask MA onto a target portion C (e.g., comprising        one or more dies) of the substrate W.

As depicted herein, the apparatus is of a transmissive type (i.e., has atransmissive mask). However, in general, it may also be of a reflectivetype, for example (with a reflective mask). Alternatively, the apparatusmay employ another kind of patterning means as an alternative to the useof a mask; examples include a programmable mirror array or LCD matrix.

The source LA (e.g., a mercury lamp or excimer laser) produces a beam ofradiation. This beam is fed into an illumination system (illuminator)IL, either directly or after having traversed conditioning means, suchas a beam expander Ex, for example. The illuminator IL may compriseadjusting means AM for setting the outer and/or inner radial extent(commonly referred to as a-outer and a-inner, respectively) of theintensity distribution in the beam. In addition, it will generallycomprise various other components, such as an integrator IN and acondenser CO. In this way, the beam PB impinging on the mask MA has adesired uniformity and intensity distribution in its cross-section.

It should be noted with regard to FIG. 20 that the source LA may bewithin the housing of the lithographic projection apparatus (as is oftenthe case when the source LA is a mercury lamp, for example), but that itmay also be remote from the lithographic projection apparatus, theradiation beam that it produces being led into the apparatus (e.g., withthe aid of suitable directing mirrors); this latter scenario is oftenthe case when the source LA is an excimer laser (e.g., based on KrF, ArFor F₂ lasing). The current invention encompasses at least both of thesescenarios.

The beam PB subsequently intercepts the mask MA, which is held on a masktable MT. Having traversed the mask MA, the beam PB passes through thelens PL, which focuses the beam PB onto a target portion C of thesubstrate W. With the aid of the second positioning means (andinterferometric measuring means IF), the substrate table WT can be movedaccurately, e.g. so as to position different target portions C in thepath of the beam PB. Similarly, the first positioning means can be usedto accurately position the mask MA with respect to the path of the beamPB, e.g., after mechanical retrieval of the mask MA from a mask library,or during a scan. In general, movement of the object tables MT, WT willbe realized with the aid of a long-stroke module (coarse positioning)and a short-stroke module (fine positioning), which are not explicitlydepicted in FIG. 20. However, in the case of a wafer stepper (as opposedto a step-and-scan tool) the mask table MT may just be connected to ashort stroke actuator, or may be fixed.

The depicted tool can be used in two different modes:

-   -   In step mode, the mask table MT is kept essentially stationary,        and an entire mask image is projected in one go (i.e., a single        “flash”) onto a target portion C. The substrate table WT is then        shifted in the x and/or y directions so that a different target        portion C can be irradiated by the beam PB;    -   In scan mode, essentially the same scenario applies, except that        a given target portion C is not exposed in a single “flash”.        Instead, the mask table MT is movable in a given direction (the        so-called “scan direction”, e.g., the y direction) with a speed        v, so that the projection beam PB is caused to scan over a mask        image; concurrently, the substrate table WT is simultaneously        moved in the same or opposite direction at a speed V=Mv, in        which M is the magnification of the lens PL (typically, M=¼ or        ⅕). In this manner, a relatively large target portion C can be        exposed, without having to compromise on resolution.

The concepts disclosed herein may simulate or mathematically model anygeneric imaging system for imaging sub wavelength features, and may beespecially useful with emerging imaging technologies capable ofproducing wavelengths of an increasingly smaller size. Emergingtechnologies already in use include EUV (extreme ultra violet)lithography that is capable of producing a 193 nm wavelength with theuse of a ArF laser, and even a 157 nm wavelength with the use of aFluorine laser. Moreover, EUV lithography is capable of producingwavelengths within a range of 20-5 nm by using a synchrotron or byhitting a material (either solid or a plasma) with high energy electronsin order to produce photons within this range. Because most materialsare absorptive within this range, illumination may be produced byreflective mirrors with a multi-stack of Molybdenum and Silicon. Themulti-stack mirror has a 40 layer pairs of Molybdenum and Silicon wherethe thickness of each layer is a quarter wavelength. Even smallerwavelengths may be produced with X-ray lithography. Typically, asynchrotron is used to produce an X-ray wavelength. Since most materialis absorptive at x-ray wavelengths, a thin piece of absorbing materialdefines where features would print (positive resist) or not print(negative resist).

While the apparatus disclosed herein may be used for imaging on asubstrate such as a silicon wafer, it shall be understood that thedisclosed concepts may be used with any type of lithographic imagingsystems, e.g., those used for imaging on substrates other than siliconwafers.

Although the present invention has been described and illustrated indetail, it is to be clearly understood that the same is by way ofillustration and example only and is not to be taken by way oflimitation, the scope of the present invention being limited only by theterms of the appended claims.

What is claimed is:
 1. A method comprising: obtaining a first modelrepresenting an imaging performance of a physical optical imaging systemconfigured to produce a pattern in or on a radiation-sensitive substrateand a physical manufacturing process utilizing the optical imagingsystem to produce the pattern in or on the substrate; simulating, by ahardware computer system, at least the imaging performance using thefirst model, the simulating generating values corresponding to a latentimage slope; defining a second model for estimating a pattern failurevalue that is other than or in addition to a line roughness value,regarding a feature to be imaged, the defining the second modelutilizing the values corresponding to the latent image slope generatedby the first model to define parameters which estimate occurrence ofpattern failure of a given feature imaged using the process, the firstmodel and/or second model defined based on measurement results of theoptical imaging system or on measurement results of an output from theprocess; and configuring the process and/or optical imaging system basedon data generated using an estimated pattern failure determined usingthe second model and/or communicating data generated based on anestimated pattern failure determined using the second model to aphysical apparatus in order to create or modify a physical configurationof the optical imaging system and/or the process.
 2. The method of claim1, wherein the second model is defined by comparing measured imagingdata to simulated imaging data generated by the first model, andselecting values for parameters contained in the second model such thatthe parameters indicate, when evaluated, whether pattern failure willoccur for the given feature based on parameter values associated withthe given feature.
 3. The method of claim 2, wherein the parameters ofthe second model include a parameter corresponding to a minimumallowable critical dimension for the given process, a parametercorresponding to a minimum allowable contrast for the given process, anda cross-term for indicating a failure condition when both the criticaldimension and the contrast are above but approximate the minimumallowable values.
 4. The method of claim 3, wherein the parameters ofthe second model further include a parameter corresponding to a minimalspace requirement between features, and a parameter corresponding to alatent image slope value associated with the minimal space requirementbetween features.
 5. The method of claim 1, wherein the latent imageslope is determined at the edge of the feature.
 6. A method comprising:obtaining a first model representing an imaging performance of aphysical optical imaging system configured to produce a pattern in or ona radiation-sensitive substrate and a physical manufacturing processutilizing the optical imaging system to produce the pattern in or on thesubstrate; simulating, by a hardware computer system, at least theimaging performance using the first model, the simulating generatingvalues corresponding to a latent image parameter; defining a secondmodel for predicting a pattern failure value that is other than or inaddition to a line roughness value, regarding a feature to be imaged,the defining the second model utilizing the values corresponding to thelatent image parameter generated by the first model to define parameterswhich predict pattern failure of a given feature imaged using theprocess, the first model and/or second model defined based onmeasurement results of the optical imaging system or on measurementresults of an output from the process; and configuring the processand/or optical imaging system based on a predicted pattern failuredetermined using the second model and/or communicating data generatedbased on a predicted pattern failure determined using the second modelto a physical apparatus in order to create or modify a physicalconfiguration of the optical imaging system and/or the process.
 7. Themethod of claim 6, wherein the second model is defined by comparingmeasured imaging data to simulated imaging data generated by the firstmodel, and selecting values for parameters contained in the second modelsuch that the parameters indicate, when evaluated, whether patternfailure is predicted to occur for the given feature based on parametervalues associated with the given feature.
 8. The method of claim 7,wherein the parameters of the second model include a parametercorresponding to a minimum allowable critical dimension for the givenprocess, a parameter corresponding to a minimum allowable contrast forthe given process, and a cross-term for indicating a failure conditionwhen both the critical dimension and the contrast are above butapproximate the minimum allowable values.
 9. The method of claim 8,wherein the parameters of the second model further include a parametercorresponding to a minimal space requirement between features, and aparameter corresponding to a latent image parameter value associatedwith the minimal space requirement between features.
 10. The method ofclaim 6, wherein the latent image parameter is determined at the edge ofthe feature.
 11. The method of claim 6, further comprising defining aline width roughness model by selecting values for parameters in theline width roughness model such that the line width roughness modelgenerates line width roughness estimates from the simulated latent imageparameter values, which correspond to measured line width roughness datawithin a predefined error tolerance.
 12. A non-transitory computerreadable storage medium having instructions stored thereon which, whenexecuted by a hardware computer system, cause the hardware computersystem to at least: obtain a first model representing an imagingperformance of a physical optical imaging system configured to produce apattern in or on a radiation-sensitive substrate and a physicalmanufacturing process utilizing the optical imaging system to producethe pattern in or on the substrate; simulate at least the imagingperformance using the first model, the simulation generating valuescorresponding to a latent image parameter; define a second model forpredicting a pattern failure value that is other than or in addition toa line roughness value, regarding a feature to be imaged, definition ofthe second model utilizing the values corresponding to the latent imageparameter generated by the first model to define parameters whichpredict pattern failure of a given feature imaged using the process, thefirst model and/or second model defined based on measurement results ofthe optical imaging system or on measurement results of an output fromthe process; and configure the process and/or optical imaging systembased on data generated using a predicted pattern failure determinedusing the second model and/or communicate data generated based on apredicted pattern failure determined using the second model to aphysical apparatus in order to create or modify a physical configurationof the optical imaging system and/or the process.
 13. The non-transitorycomputer readable storage medium of claim 12, wherein the second modelis defined by comparing experimental imaging data to simulated imagingdata generated by the first model, and the instructions are furtherconfigured to cause the hardware computer system to implement selectionof values for parameters contained in the second model such that theparameters indicate, when evaluated, whether pattern failure ispredicted to occur for the given feature based on parameter valuesassociated with the given feature.
 14. The non-transitory computerreadable storage medium of claim 12, wherein the parameters of thesecond model include a parameter corresponding to a minimum allowablecritical dimension for the given process, a parameter corresponding to aminimum allowable contrast for the given process, and a cross-term forindicating a failure condition when both the critical dimension and thecontrast are above but approximate the minimum allowable values.
 15. Thenon-transitory computer readable storage medium of claim 14, wherein theparameters of the second model further include a parameter correspondingto a minimal space requirement between features, and a parametercorresponding to a latent image parameter value associated with theminimal space requirement between features.
 16. The non-transitorycomputer readable storage medium of claim 12, wherein the latent imageparameter is determined at the edge of the feature.
 17. Thenon-transitory computer readable storage medium of claim 12, wherein theinstructions are further configured to cause the hardware computersystem to implement definition of a line width roughness model byselecting values for parameters in the line width roughness model suchthat the line width roughness model generates line width roughnessestimates from the simulated latent image parameter values, whichcorrespond to measured line width roughness data within a predefinederror tolerance.
 18. A method comprising: obtaining a model forestimating a pattern failure regarding a feature to be imaged by aphysical optical imaging system configured to produce a pattern in or ona radiation-sensitive substrate as part of a physical manufacturingprocess utilizing the optical imaging system to produce the pattern inor on the substrate, the model comprising parameters to estimateoccurrence of pattern failure of a given feature imaged using theprocess, the parameters of the model including a parameter representinga minimum allowable critical dimension for the given process and aparameter representing a minimum allowable contrast for the givenprocess; determining, by a hardware computer system, an estimate of apattern failure of the optical imaging system and/or the process usingthe model; and configuring the process and/or optical imaging systembased on data generated using the estimate of the pattern failure and/orcommunicating data generated based on the estimate of the patternfailure to a physical apparatus in order to create or modify a physicalconfiguration of the optical imaging system and/or the process.
 19. Themethod of claim 18, wherein the parameters of the model further comprisea cross-term for indicating a failure condition when both the criticaldimension and the contrast are above but approximate the minimumallowable values.
 20. The method of claim 19, wherein the parameters ofthe model further include a parameter corresponding to a minimal spacerequirement between features and a parameter correspond to a latentimage parameter value associated with the minimal space requirementbetween features.
 21. A non-transitory computer readable storage mediumhaving instructions stored thereon which, when executed by a hardwarecomputer system, cause the hardware computer system to at least: obtaina model for estimating a pattern failure regarding a feature to beimaged by a physical optical imaging system configured to produce apattern in or on a radiation-sensitive substrate as part of a physicalmanufacturing process utilizing the optical imaging system to producethe pattern in or on the substrate, the model comprising parameters toestimate occurrence of pattern failure of a given feature imaged usingthe process, the parameters of the model including a parameterrepresenting a minimum allowable critical dimension for the givenprocess and a parameter representing a minimum allowable contrast forthe given process; determine an estimate of a pattern failure of theoptical imaging system and/or the process using the model; and configurethe process and/or optical imaging system based on data generated usingthe estimate of the pattern failure and/or communicate data generatedbased on the estimate of the pattern failure to a physical apparatus inorder to create or modify a physical configuration of the opticalimaging system and/or the process.
 22. The non-transitory computerreadable storage medium of claim 21, wherein the parameters of the modelfurther comprise a cross-term for indicating a failure condition whenboth the critical dimension and the contrast are above but approximatethe minimum allowable values.
 23. The non-transitory computer readablestorage medium of claim 22, wherein the parameters of the model furtherinclude a parameter corresponding to a minimal space requirement betweenfeatures and a parameter correspond to a latent image parameter valueassociated with the minimal space requirement between features.